# Precedence graph

A precedence graph, also named conflict graph and serializability graph, is used in the context of concurrency control in databases.

The precedence graph for a schedule S contains:

• A node for each committed transaction in S
• An arc from Ti to Tj if an action of Ti precedes and conflicts with one of Tj's actions.

## Precedence graph examples

### Example 1

${\displaystyle S={\begin{bmatrix}T1&T2\\R(A)&R(A)\\A=A*5&R(B)\\W(A)&B=B+A\\R(B)&W(B)\\B=B*10&\\W(B)&\\\end{bmatrix}}}$

### Example 2

${\displaystyle D=R1(A)}$ ${\displaystyle R2(B)}$ ${\displaystyle W2(A)}$ ${\displaystyle Com.2}$ ${\displaystyle W1(A)}$ ${\displaystyle Com.1}$ ${\displaystyle W3(A)}$ ${\displaystyle Com.3}$

A precedence graph of the schedule D, with 3 transactions. As there is a cycle (of length 2; with two edges) through the committed transactions T1 and T2, this schedule (history) is not Conflict serializable. Notice, that the commit of Transaction 2 does not have any meaning regarding the creation of a precedence graph.

## Testing Serializability with Precedence Graph

Algorithm to test Conflict Serializability of a Schedule S along with an example schedule.

${\displaystyle S={\begin{bmatrix}T1&T2&T3\\R(A)&&\\&W(A)&\\&Com.&\\W(A)&&\\Com.&&\\&&W(A)\\&&Com.\\\end{bmatrix}}}$

or

${\displaystyle S=R1(A)}$ ${\displaystyle W2(A)}$ ${\displaystyle Com.2}$ ${\displaystyle W1(A)}$ ${\displaystyle Com.1}$ ${\displaystyle W3(A)}$ ${\displaystyle Com.3}$

1. For each transaction Tx participating in schedule S, create a node labeled Ti in the precedence graph. Thus the precedence graph contains T1, T2, T3.
2. For each case in S where Tj executes a read_item(X) after Ti executes a write_item(X), create an edge (Ti → Tj) in the precedence graph. This occurs nowhere in the above example, as there is no read after write.
3. For each case in S where Tj executes a write_item(X) after Ti executes a read_item(X), create an edge (Ti → Tj) in the precedence graph. This results in a directed edge from T1 to T2 (as T1 has R(A) before T2 having W(A)).
4. For each case in S where Tj executes a write_item(X) after Ti executes a write_item(X), create an edge (Ti → Tj) in the precedence graph. This results in directed edges from T2 to T1, T2 to T3 and T1 to T3.
5. The schedule S is serializable if and only if the precedence graph has no cycles. As T1 and T2 constitute a cycle, the above example is not (conflict) serializable.